Section 2.2 Calculus of Vector Functions

2.2 Calculus of Vector Functions

In this section, we calculus of vector functions, i.e. the derivatives of vector functions and integrations of vector functions. They are essential the same as functions of one variable. The only difference is that vector functions have components.

Definition: 

(a) The derivative of a vector function $\overrightarrow{r(t)}$ at $t=a$ is defined as

$$\overrightarrow{r^{\prime }(a)}=\underset{h\to 0}{lim}\frac{\overrightarrow{r(a+h})-\overrightarrow{r(a)}}{h}$$

if the limit exits. The derivative vector function is denoted as $\overrightarrow{r^{\prime }(t)}$.

(b) The tangent line to the curve $\overrightarrow{r(t)}=<f(t),g(t),h(t)>$ at the point $P=(f(a),g(a),h(a))$ is defined to be the line through $P$ parallel to the tangent vector $\overrightarrow{r^{\prime }(a)}$. Hence we sometimes call $\overrightarrow{r^{\prime }(t)}$ the tangent vector function of $\overrightarrow{r(t)}$.

(c) We will also have occasion to consider the unit tangent vector, which is

$$\overrightarrow{T(t)}=\frac{\overrightarrow{r^{\prime }(t)}}{||\overrightarrow{r^{\prime }(t)}||}.$$

Theorem:

$\overrightarrow{r(t)}=<f(t),g(t),h(t)>=f(t)\overrightarrow{i}+g(t)\overrightarrow{j}+h(t)\overrightarrow{k}$ be a vector function such that $f(t)$, $g(t)$ and $h(t)$ are differential functions then 

$$\overrightarrow{r^{\prime }(t)}=<f^{\prime }(t),g^{\prime }(t),h^{\prime }(t)>=f^{\prime }(t)\overrightarrow{i}+g^{\prime }(t)\overrightarrow{j}+h^{\prime }(t)\overrightarrow{k}.$$

Example 1: Find $\overrightarrow{r^{\prime }(t)}$ where $\overrightarrow{r(t)}=<t^2+1,\text{sin}(t),e^{2t}>$. Find the unit tangent vector at $t=0$.

Exercise 1: Find $\overrightarrow{r^{\prime }(t)}$ where $\overrightarrow{r(t)}=<3t^2-t,e^{3t},\text{cos}(t)>$. Find the unit tangent vector at $t=0$.

Example 2: Find $\overrightarrow{r^{\prime }(t)}$ where $\overrightarrow{r(t)}=<\text{ln}(t+1),{\text{sin}}^{-1}(t),t{e}^{2t}>$. Find the unit tangent vector at $t=0$.

Exercise 2: Find $\overrightarrow{r^{\prime }(t)}$ where $\overrightarrow{r(t)}=<\text{ln}(1+t^2),{\text{tan}}^{-1}(t),\text{cos}(t)e^t>$. Find the unit tangent vector at $t=0$.

Fact:  The second derivative of a vector function $\overrightarrow{r(t)}$ is the derivative of $\overrightarrow{r^{\prime }(t)}$.

Example 3: $\overrightarrow{r(t)}=<\text{sin}(t),\text{cos}(t),t^2>$. Find $\overrightarrow{T(0)}$, $\overrightarrow{r^{″}(0)}$, $\overrightarrow{r^{\prime }(0)}\cdot \overrightarrow{r^{″}(0)}$ and $\overrightarrow{r^{\prime }(0)}\times \overrightarrow{r^{″}(0)}$.

Exercise 3: $\overrightarrow{r(t)}=<t^2+1,\text{cos}(t),\text{sin}(t)>$. Find $\overrightarrow{T(0)}$, $\overrightarrow{r^{″}(0)}$, $\overrightarrow{r^{\prime }(0)}\cdot \overrightarrow{r^{″}(0)}$ and $\overrightarrow{r^{\prime }(0)}\times \overrightarrow{r^{″}(0)}$.

Theorem: Properties of the Derivative of Vector-Valued Functions 

Let $\overrightarrow{r(t)}$ and $\overrightarrow{u(t)}$ be differentiable vector functions of $t$, let $f(t)$ be a differentiable function of $t$, and let $c$ be a scalar.

$$(i)\frac{d}{dt}[c\overrightarrow{r(t)}]=c\frac{d}{dt}\overrightarrow{r^{\prime }(t)}\text{ Scalar multiple}$$ $$(ii)\frac{d}{dt}[\overrightarrow{r(t)}\pm \overrightarrow{u(t)}]=\frac{d}{dt}[\overrightarrow{r(t)}]+\frac{d}{dt}[\overrightarrow{u(t)}]\text{ Sum and difference}$$ $$(iii)\frac{d}{dt}[f(t)\overrightarrow{r(t)}]=[\frac{d}{dt}f(t)]\overrightarrow{r(t)}\pm f(t)[\frac{d}{dt}\overrightarrow{r(t)}]\text{ Scalar product}$$ $$(iv)\frac{d}{dt}[\overrightarrow{r(t)}\cdot \overrightarrow{u(t)}]=[\frac{d}{dt}\overrightarrow{r(t)}]\cdot \overrightarrow{u(t)}+\overrightarrow{r(t)}\cdot [\frac{d}{dt}\overrightarrow{u(t)}]\text{ Dot product}$$ $$(v)\frac{d}{dt}[\overrightarrow{r(t)}\times \overrightarrow{u(t)}]=[\frac{d}{dt}\overrightarrow{r(t)}]\times \overrightarrow{u(t)}+\overrightarrow{r(t)}\times [\frac{d}{dt}\overrightarrow{u(t)}]\text{ Cross product}$$ $$(vi)\frac{d}{dt}[\overrightarrow{r(f(t))}]=[\frac{d}{dt}\overrightarrow{r(f(t))}]\frac{d}{dt}f(t)\text{ Chain rule}$$ $$(vii)\text{If }||\overrightarrow{r(t)}||=c\text{Then }\overrightarrow{r(t)}\cdot \overrightarrow{r^{\prime }(t)}=0$$

Example 4: $\overrightarrow{r(t)}=<\text{sin}(t),\text{cos}(t),t^2>$ and $\overrightarrow{u(t)}=<\sqrt{t},e^t,\text{tan}(t)>.$ Find $\frac{d}{dt}[\overrightarrow{r(t)}\cdot \overrightarrow{u(t)}]$ and $\frac{d}{dt}[\overrightarrow{r(t)}\times \overrightarrow{u(t)}]$.

Exercise 4: $\overrightarrow{r(t)}=<t^2+1,\text{cos}(t),\text{sin}(t)>$ and $\overrightarrow{u(t)}=<t,\text{sec}(t),e^t>.$ Find $\frac{d}{dt}[\overrightarrow{r(t)}\cdot \overrightarrow{u(t)}]$ and $\frac{d}{dt}[\overrightarrow{r(t)}\times \overrightarrow{u(t)}]$.

Definition

Let $f(t)$, $g(t)$ and $h(t)$ be integrable real-valued functions over the closed interval $[a,b]$.

(a) The indefinite integral of a vector-valued function $\overrightarrow{r(t)}=<f(t),g(t),h(t)>$ is $$\int \overrightarrow{r}(t)dt=<\int f(t)dt,\int g(t)dt,\int h(t)dt>.$$

(b) The definite integral of a vector-valued function $\overrightarrow{r(t)}=<f(t),g(t),h(t)>$ is $${\int }_a^b\overrightarrow{r}(t)dt=<{\int }_a^{b}f(t)dt,{\int }_a^{b}g(t)dt,{\int }_a^{b}h(t)dt>$$ $$=<F(b)-F(a),G(b)-G(a),H(b)-H(a)>$$ where $F^{\prime }(t)=f(t)$, $G^{\prime }(t)=g(t)$ and $H^{\prime }(t)=h(t)$. 

Example 5: $\overrightarrow{r(t)}=<\text{sin}(t),\text{cos}(t),t^2>$. Find $\int \overrightarrow{r(t)}dt$ and ${\int }_0^1\overrightarrow{r(t)}dt$.

Exercise 5: $\overrightarrow{r(t)}=<t^2+1,\text{cos}(t),\text{sin}(t)>$. Find $\int \overrightarrow{r(t)}dt$ and ${\int }_0^1\overrightarrow{r(t)}dt$.

Example 6: Find $\overrightarrow{r(t)}$ where $\overrightarrow{r^{\prime }(t)}=<\text{sin}(t),\text{cos}(t),e^{2t}>$ and $\overrightarrow{r(0)}=<1,1,2>$.

Exercise 6: Find $\overrightarrow{r(t)}$ where $\overrightarrow{r^{\prime }(t)}=<2t,e^{3t},\text{sin}(t)>$ and $\overrightarrow{r(0)}=<0,1,1>$.

Example 7: Find the parameter equations of the tangent line of the given vector function at the point $(1,0,1)$

$$\overrightarrow{r(t)}=<e^{-t}\text{cos}(t),e^{-t}\text{sin}(t),e^{2t}>.$$

Example 8: At what point two curves, $\overrightarrow{r(t)}=<2+t,t^2,1-2t>$ and $\overrightarrow{u(s)}=<s,2-s,s-1>$ intersect? Find the angle of the intersection.

Group work:

1. Find the parameter equations of the tangent line of the given vector function at the point $(0,0,1)$

$$\overrightarrow{r(t)}=<\text{ln}(t+1),t\text{cos}(3t),5^t>.$$

2. At what point two curves, $\overrightarrow{r(t)}=<t,1-t,3+t^2>$ and $\overrightarrow{u(s)}=<3-s,s-2,s^2>$ intersect? Find the angle of the intersection.

3. A particle travels along the path of a helix with the equation $\overrightarrow{r(t)}=<\text{cos}(t),\text{sin}(t),t>$.

(a) Velocity of the particle at any time.

(b) Speed of the particle at any time.

(c) Acceleration of the particle at any time.

(d) Find the unit tangent vector for the helix.